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Jackson and smoothness theorems for Freud weights in \(L_ p (0<p\leq\infty)\). (English) Zbl 0867.41010
The authors study Jackson, realization and converse theorems for Freud weights in \(L_p\) spaces, \(0<p \leq\infty\). The method adopted here even for the case \(1\leq p\leq \infty\) for Jackson theorems for Freud weights avoids the use of the deep properties of orthogonal polynomials. Some properties of the modulus of smoothness in the \(L_p\) space for \(0<p\leq \infty\) have been given involving realization functionals. The technique of first approximating by a spline and then by a polynomial as followed here seems to be new in the context and is of some intrinsic interest. Various other interesting results, such as some Marchaud-type inequalities, are also given.

41A10 Approximation by polynomials
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis